Tamara J. answered • 03/28/13

Math Tutoring - Algebra and Calculus (all levels)

Let's summarize the 2 trip plans:

Trip A: # of nights = 3

pairs of theme park tickets = 2

total cost of trip = $415

Trip B: # of nights = 5

pairs of theme park tickets = 4

total cost of trip = $725

Questions 1 and 2 ask you to find an equation for trip A and trip B, respectively, given the following:

x = cost of hotel per night

y = cost per pair of theme park tickets

(1.) Total cost of hotel stay is the cost of the hotel per night (x) multiplied by the number of nights (3), which is to say that the total cost of the hotel stay for 3 nights is equal to 3x. The total cost of the pairs of theme park tickets is the cost per pair of theme park tickets (y) multiplied by the number of pairs of theme park tickets offered (2), which means that the total cost of the theme park tickets for 2 pairs is equal to 2y. Since the total cost of the trip plan is given to be $415, then the equation representing the plan for trip A is as follows:

3x + 2y = 415

(2.) The equation for trip B is obtained by similar means as is it was for trip A. Given that the total cost of trip B is $725, with a hotel stay for a total of 5 nights and 4 pairs of theme park tickets are offered, we arrive at the following equation:

5x + 4y = 725

(3.) The system of equations, thus, contains the following equation:

3x + 2y = 415

5x + 4y = 725

We can solve for the system using either the substitution method or the method of elimination. I find the method of elimination easier, so I will demonstrate how to solve for this system using this method by first multiplying the first equation by -2 to eliminate the y variable, which allows us to solve for the x variable first.

-2(3x + 2y = 415) ==> -6x - 4y = -830

Combine this manipulated equation to the second original equation:

-6x - 4y = -830

+ 5x + 4y = 725

_____________________

-1x + 0 = -105

-1x = -105

Divide both sides of the equation by -1 to solve for x:

-1x/-1 = -105/-1

x = 105

Solve for y by plugging in this value for x into one of the original equations:

3x + 2y = 415

3(105) + 2y = 415

315 + 2y = 415

2y = 100

y = 50

Thus, the cost of the hotel stay per night (x) is $105 and the cost of each pair of theme park tickets (y) is $50.